Impact of Channel Estimation Error on Adaptive Modulation ... - UMA
716
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 5, MAY 2004
Impact of Channel Estimation Error on Adaptive Modulation Performance in Flat Fading José F. Paris, M. Carmen Aguayo-Torres, and José T. Entrambasaguas
Abstract—Presented here is an approach to analyzing the effect of imperfect channel estimation on adaptive modulation. The sensitivity of the main performance parameters to short-term and long-term estimation error sources is summarized in a set of new formulas that are either closed-form expressions or simple to compute numerically. Index Terms—Adaptive flat-fading channels.
modulation,
channel
the adaptive modulation scheme and the channel estimation subsystem considered here. In Section III, close approximations are found for the cutoff SNR to simplify the performance analysis carried out in Section IV. Finally, conclusions about the presented approach are provided in Section V.
estimation,
I. INTRODUCTION
A
DAPTIVE modulation is a promising technique that is being progressively incorporated into wireless communication systems [1]–[6]. One important premise for adaptive modulation performance is the availability of an accurate knowledge of the channel fading. In a likely scenario, the time-varying frequency-flat slow fading consists of short-term variations, due to multipath propagation, and long-term variations as a consequence of path loss/shadowing. Typically, short-term variations are modeled by a Rayleigh distribution, and long-term variations by a log-normal distribution. Information concerning both phenomena is needed to implement adaptive modulation schemes. For this task, the instantaneous signal-to-noise ratio (SNR) as the short-term descriptor, and the average SNR as the long-term descriptor, are usually considered. Pilot symbols can be inserted into the data stream to obtain this information. Finally, the transmitter is dynamically configured according to these estimates to optimize the system spectral efficiency. Any uncertainty about the channel fading descriptors may cause significant degradation of adaptive modulation performance. Several papers have addressed the usage of imperfect channel estimates in adaptive modulation [1], [7], [8]. In [1], a simple approximation to evaluate the effect of delayed instantaneous SNR estimates on the bit-error rate (BER) is presented. The characterization of the channel fading variations is considered in [7], to design effective adaptive modulation schemes for systems using outdated fading estimates. Finally, in [8], the impact of the estimation delay on the BER is analyzed for adaptive quadrature amplitude modulation (AQAM). The motivation of this letter is providing an approach for the analysis of the combined effect of the different error sources related to the channel estimation for AQAM. The remainder of this letter is organized as follows. Section II briefly describes Paper approved by M.-S. Alouini, the Editor for Modulation and Diversity Systems of the IEEE Communications Society. Manuscript received April 25, 2003; revised September 9, 2003 and October 19, 2003. This work was supported in part by the Spain CICYT under project TIC2003-07819. The authors are with the Department of Ingeniería de Comunicaciones, University of Málaga, Málaga E-29071, Spain (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.826254
II. SYSTEM MODEL Fig. 1 shows a block diagram of the adaptive modulation system. It is assumed flat fading, thus, the channel model consists of the baseband complex envelope plus additive white Gaussian noise (AWGN) . The received , where power follows a multiplicative model is the power fluctuation due to path loss/shadowing, and is the power fluctuation due to multipath with . Although is time dependent, its variation is assumed very slow, compared with the temporal variations of characterized by the Doppler spread . From the point of view of exhibits Rayleigh fading short-term temporal variations, with average power gain . When long-term variations are also considered, is modeled as a random variable (RV) that is widely accepted as being log-normal. The aforementioned random processes are ergodic, stationary, and statistically independent, thus, temporal references can be dropped for simplicity of notation. At the transmitter, fixed pilot symbols are periodically inserted into the data symbols. Both types of symbols have the same average power1 and are assembled into frames that condata symbols. Among sist of one pilot symbol followed by all possible AQAM schemes, continuous-rate continuous-power adaptation with instantaneous BER constraint is chosen, as in [8], for the analysis performed in next sections. This scheme preserves all degrees of freedom concerning rate-power adaptation, while achieving the target BER in any channel state. The (bits/symbol) and power adaptation rate adaptation that maximize the spectral efficiency subject to the instantaare [2] neous BER constraint (1) where , the average transmitted power, and is the unit-step function, i.e., power transmission is disabled below the cutoff SNR . The instantaneous BER is kept constant above the cutoff SNR, thus, it is fixed to a target value
(2) 1The
choice of equal data and pilot symbol power is not necessarily optimal.
0090-6778/04$20.00 © 2004 IEEE
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 5, MAY 2004
Fig. 1.
717
AQAM system model, including the channel estimation subsystem.
At the receiver, channel estimation is performed by the symbol pilots inserted at the transmitter [9], [10]. As shown in Fig. 1, the channel estimation subsystem required by the AQAM algorithm can be split into two parts, which provide estimates for the instantaneous SNR ( ) and average SNR ( ). The estimate can be obtained from a finite-impulse response (FIR) filtered version of samples of the channel comfollowed by a squared norm operation.2 Folplex envelope lowing the analysis presented in [10], the conditional probability is then derived, assuming the density function (PDF) filter coefficients are chosen under the minimum mean-square error (MMSE) criterion
with rameter
Rayleigh fading can be determined by solving [2]
(5)
where and are the normalized instancan taneous SNR and cutoff SNR, respectively. Although be bounded and numerically calculated [13], it is not possible to find an exact closed-form solution. The cutoff SNR can be . An initial approximation expressed as , within 1 dB for the usual range of for the cutoff function , is given in [14] by values of and
(3) the zeroth-order modified Bessel function. The parepresents the MMSE of
(6)
where tighter closed-form approximations are also provided. The function is strictly monotonically increasing, and , then . when (4) and depends on , the number of filter taps , and the normalized frame interval ( is the symbol interval). Note that is the normalized covariance matrix given by , is the -dimensional normalized covariance row vector where , and is the zeroth-order Bessel function. With regard to the estimate of the average SNR, [12] provides a simple model to characterize its statistics for different average power measurement methods. More specifically, the relative error is nearly Gauss distributed, with its mean easily driven to zero and its standard deviation within the 2–4 dB range. III. APPROXIMATIONS FOR THE CUTOFF SNR All adaptive modulation performance parameters depend on the cutoff SNR. For each given value of , the cutoff SNR for 2This estimator could be improved by considering the bias in the squared norm channel gain [11].
IV. PERFORMANCE ANALYSIS In this section, several expressions are obtained to evaluate the impact of the imperfect channel estimates and on the more relevant AQAM performance parameters: BER, spectral efficiency, average transmitted power, and outage probability. For each parameter, the analysis is split into two steps. First, the short-term error only is considered, and later the long-term error is included. Finally, numerical and simulation results show the relevance of the different error sources. A. Averaged BER Analytical results for the BER, shown in (7)–(8) at the bottom of the next page, are derived as follows. Let us suppose that and . The instantaneous BER is no longer kept constant and equal to above the cutoff SNR. According to Fig. 1, two different impairments can be identified: rate-power selection is performed by using the noisy estimate and, when the transmitter starts to send data with this selection, the channel
718
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 5, MAY 2004
has evolved to a new state . Therefore, from (2), the instantaneous BER is now given by
(9) where and represent the noisy and delayed instantaneous SNR ( ) and the current instantaneous SNR ( ), respectively. Note that when because there is no data transmission. From the standpoint of performance evaluation, the averaged BER above the cutoff SNR is used to quantify the variation of the instantaneous BER
(10) Consistently, when channel estimation is perfect, then . Evaluation of (10) only requires the identification of the , since can be derived from it. In [8, joint PDF p. 136, eq. (41)], the conditional PDF is obtained as a function of two parameters: the adaptation delay normalized , and the channel gain correlation to the Doppler shift coefficient . Hence, the joint PDF can be obtained if the following conditional probability chain is regarded:
When the long-term error is included in the analysis, i.e., , the cutoff SNR becomes
(13) and expectation over must be applied to (12a). Then, after , we obtain considering the change of variable (7), where the former is redefined as
(14) and and as previously defined in (12b). Integration over and in (7) can be accurately approximated by the Gauss–Laguerre and Gauss–Hermite quadrature formulas, respectively. and are the order By this method, (8) is obtained, where and the zeros of the Laguerre and Hermite polynomials, of such polynomials, and and the associated weight factors [16, p. 223–224]. B. Spectral Efficiency, Average Transmitted Power, and Outage Probability A similar analysis can be carried out for these performance parameters. By integrating (11) with [15, p. 1182, eq. (109)], it is exponential distributed is shown that the marginal PDF with mean . Consequently, considering (1), it is straightforward to evaluate how the short-term error affects the , average transmitted power , and spectral efficiency3 outage probability
(15)
(11) Substituting (3) and [8, p. 136, eq. (41)] into (11), then (11) in (10), using twice the Laplace transform given in [15, p. 1182, eq. (109)] and applying the change of variable allows us to obtain
(12a) (12b)
(16)
(17) is the first-order exponential-integral function where classically defined as in [17, p. xxxiv]. 3The spectral efficiency penalty factor (L pilot symbols is not included.
0 1)=L caused by the insertion of
(7) (8)
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 5, MAY 2004
719
Fig. 2. Impact of channel estimation error on the AQAM system performance parameters as a function of the MSE dB, for different channel conditions and dB with and T . Simulation results are also shown for M 1, 2, 4, 8, and values of the long-term error standard deviation 16 filter taps. (a) Averaged BER in terms of the normalized adaptation delay for any ; . (b) Relative variation of the spectral efficiency S =S . (d) Relative variation of the outage probability = . (c) Relative variation of the average transmitted power
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 5, MAY 2004
Impact of Channel Estimation Error on Adaptive Modulation Performance in Flat Fading José F. Paris, M. Carmen Aguayo-Torres, and José T. Entrambasaguas
Abstract—Presented here is an approach to analyzing the effect of imperfect channel estimation on adaptive modulation. The sensitivity of the main performance parameters to short-term and long-term estimation error sources is summarized in a set of new formulas that are either closed-form expressions or simple to compute numerically. Index Terms—Adaptive flat-fading channels.
modulation,
channel
the adaptive modulation scheme and the channel estimation subsystem considered here. In Section III, close approximations are found for the cutoff SNR to simplify the performance analysis carried out in Section IV. Finally, conclusions about the presented approach are provided in Section V.
estimation,
I. INTRODUCTION
A
DAPTIVE modulation is a promising technique that is being progressively incorporated into wireless communication systems [1]–[6]. One important premise for adaptive modulation performance is the availability of an accurate knowledge of the channel fading. In a likely scenario, the time-varying frequency-flat slow fading consists of short-term variations, due to multipath propagation, and long-term variations as a consequence of path loss/shadowing. Typically, short-term variations are modeled by a Rayleigh distribution, and long-term variations by a log-normal distribution. Information concerning both phenomena is needed to implement adaptive modulation schemes. For this task, the instantaneous signal-to-noise ratio (SNR) as the short-term descriptor, and the average SNR as the long-term descriptor, are usually considered. Pilot symbols can be inserted into the data stream to obtain this information. Finally, the transmitter is dynamically configured according to these estimates to optimize the system spectral efficiency. Any uncertainty about the channel fading descriptors may cause significant degradation of adaptive modulation performance. Several papers have addressed the usage of imperfect channel estimates in adaptive modulation [1], [7], [8]. In [1], a simple approximation to evaluate the effect of delayed instantaneous SNR estimates on the bit-error rate (BER) is presented. The characterization of the channel fading variations is considered in [7], to design effective adaptive modulation schemes for systems using outdated fading estimates. Finally, in [8], the impact of the estimation delay on the BER is analyzed for adaptive quadrature amplitude modulation (AQAM). The motivation of this letter is providing an approach for the analysis of the combined effect of the different error sources related to the channel estimation for AQAM. The remainder of this letter is organized as follows. Section II briefly describes Paper approved by M.-S. Alouini, the Editor for Modulation and Diversity Systems of the IEEE Communications Society. Manuscript received April 25, 2003; revised September 9, 2003 and October 19, 2003. This work was supported in part by the Spain CICYT under project TIC2003-07819. The authors are with the Department of Ingeniería de Comunicaciones, University of Málaga, Málaga E-29071, Spain (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.826254
II. SYSTEM MODEL Fig. 1 shows a block diagram of the adaptive modulation system. It is assumed flat fading, thus, the channel model consists of the baseband complex envelope plus additive white Gaussian noise (AWGN) . The received , where power follows a multiplicative model is the power fluctuation due to path loss/shadowing, and is the power fluctuation due to multipath with . Although is time dependent, its variation is assumed very slow, compared with the temporal variations of characterized by the Doppler spread . From the point of view of exhibits Rayleigh fading short-term temporal variations, with average power gain . When long-term variations are also considered, is modeled as a random variable (RV) that is widely accepted as being log-normal. The aforementioned random processes are ergodic, stationary, and statistically independent, thus, temporal references can be dropped for simplicity of notation. At the transmitter, fixed pilot symbols are periodically inserted into the data symbols. Both types of symbols have the same average power1 and are assembled into frames that condata symbols. Among sist of one pilot symbol followed by all possible AQAM schemes, continuous-rate continuous-power adaptation with instantaneous BER constraint is chosen, as in [8], for the analysis performed in next sections. This scheme preserves all degrees of freedom concerning rate-power adaptation, while achieving the target BER in any channel state. The (bits/symbol) and power adaptation rate adaptation that maximize the spectral efficiency subject to the instantaare [2] neous BER constraint (1) where , the average transmitted power, and is the unit-step function, i.e., power transmission is disabled below the cutoff SNR . The instantaneous BER is kept constant above the cutoff SNR, thus, it is fixed to a target value
(2) 1The
choice of equal data and pilot symbol power is not necessarily optimal.
0090-6778/04$20.00 © 2004 IEEE
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 5, MAY 2004
Fig. 1.
717
AQAM system model, including the channel estimation subsystem.
At the receiver, channel estimation is performed by the symbol pilots inserted at the transmitter [9], [10]. As shown in Fig. 1, the channel estimation subsystem required by the AQAM algorithm can be split into two parts, which provide estimates for the instantaneous SNR ( ) and average SNR ( ). The estimate can be obtained from a finite-impulse response (FIR) filtered version of samples of the channel comfollowed by a squared norm operation.2 Folplex envelope lowing the analysis presented in [10], the conditional probability is then derived, assuming the density function (PDF) filter coefficients are chosen under the minimum mean-square error (MMSE) criterion
with rameter
Rayleigh fading can be determined by solving [2]
(5)
where and are the normalized instancan taneous SNR and cutoff SNR, respectively. Although be bounded and numerically calculated [13], it is not possible to find an exact closed-form solution. The cutoff SNR can be . An initial approximation expressed as , within 1 dB for the usual range of for the cutoff function , is given in [14] by values of and
(3) the zeroth-order modified Bessel function. The parepresents the MMSE of
(6)
where tighter closed-form approximations are also provided. The function is strictly monotonically increasing, and , then . when (4) and depends on , the number of filter taps , and the normalized frame interval ( is the symbol interval). Note that is the normalized covariance matrix given by , is the -dimensional normalized covariance row vector where , and is the zeroth-order Bessel function. With regard to the estimate of the average SNR, [12] provides a simple model to characterize its statistics for different average power measurement methods. More specifically, the relative error is nearly Gauss distributed, with its mean easily driven to zero and its standard deviation within the 2–4 dB range. III. APPROXIMATIONS FOR THE CUTOFF SNR All adaptive modulation performance parameters depend on the cutoff SNR. For each given value of , the cutoff SNR for 2This estimator could be improved by considering the bias in the squared norm channel gain [11].
IV. PERFORMANCE ANALYSIS In this section, several expressions are obtained to evaluate the impact of the imperfect channel estimates and on the more relevant AQAM performance parameters: BER, spectral efficiency, average transmitted power, and outage probability. For each parameter, the analysis is split into two steps. First, the short-term error only is considered, and later the long-term error is included. Finally, numerical and simulation results show the relevance of the different error sources. A. Averaged BER Analytical results for the BER, shown in (7)–(8) at the bottom of the next page, are derived as follows. Let us suppose that and . The instantaneous BER is no longer kept constant and equal to above the cutoff SNR. According to Fig. 1, two different impairments can be identified: rate-power selection is performed by using the noisy estimate and, when the transmitter starts to send data with this selection, the channel
718
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 5, MAY 2004
has evolved to a new state . Therefore, from (2), the instantaneous BER is now given by
(9) where and represent the noisy and delayed instantaneous SNR ( ) and the current instantaneous SNR ( ), respectively. Note that when because there is no data transmission. From the standpoint of performance evaluation, the averaged BER above the cutoff SNR is used to quantify the variation of the instantaneous BER
(10) Consistently, when channel estimation is perfect, then . Evaluation of (10) only requires the identification of the , since can be derived from it. In [8, joint PDF p. 136, eq. (41)], the conditional PDF is obtained as a function of two parameters: the adaptation delay normalized , and the channel gain correlation to the Doppler shift coefficient . Hence, the joint PDF can be obtained if the following conditional probability chain is regarded:
When the long-term error is included in the analysis, i.e., , the cutoff SNR becomes
(13) and expectation over must be applied to (12a). Then, after , we obtain considering the change of variable (7), where the former is redefined as
(14) and and as previously defined in (12b). Integration over and in (7) can be accurately approximated by the Gauss–Laguerre and Gauss–Hermite quadrature formulas, respectively. and are the order By this method, (8) is obtained, where and the zeros of the Laguerre and Hermite polynomials, of such polynomials, and and the associated weight factors [16, p. 223–224]. B. Spectral Efficiency, Average Transmitted Power, and Outage Probability A similar analysis can be carried out for these performance parameters. By integrating (11) with [15, p. 1182, eq. (109)], it is exponential distributed is shown that the marginal PDF with mean . Consequently, considering (1), it is straightforward to evaluate how the short-term error affects the , average transmitted power , and spectral efficiency3 outage probability
(15)
(11) Substituting (3) and [8, p. 136, eq. (41)] into (11), then (11) in (10), using twice the Laplace transform given in [15, p. 1182, eq. (109)] and applying the change of variable allows us to obtain
(12a) (12b)
(16)
(17) is the first-order exponential-integral function where classically defined as in [17, p. xxxiv]. 3The spectral efficiency penalty factor (L pilot symbols is not included.
0 1)=L caused by the insertion of
(7) (8)
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 5, MAY 2004
719
Fig. 2. Impact of channel estimation error on the AQAM system performance parameters as a function of the MSE dB, for different channel conditions and dB with and T . Simulation results are also shown for M 1, 2, 4, 8, and values of the long-term error standard deviation 16 filter taps. (a) Averaged BER in terms of the normalized adaptation delay for any ; . (b) Relative variation of the spectral efficiency S =S . (d) Relative variation of the outage probability = . (c) Relative variation of the average transmitted power
